Synthesis of Discrete-time Control Barrier Functions for Polynomial Systems Based on Sum-of-Squares Programming

Discrete-time Control Barrier Functions (DTCBFs) are commonly utilized in the literature as a powerful tool for synthesizing control policies that guarantee safety of discrete-time dynamical systems. However, the systematic synthesis of DTCBFs in a computationally efficient way is at present an important open problem. This article first proposes a novel alternating-descent approach based on Sum-of-Squares programming to synthesize quadratic DTCBFs and corresponding polynomial control policies for discrete-time control-affine polynomial systems with input constraints and semi-algebraic safe sets. Subsequently, two distinct approaches are introduced to extend the proposed method to the synthesis of higher-degree polynomial DTCBFs. To demonstrate its efficacy, we apply the proposed method to numerical case studies.